Question

write an expression to describe the sequence below, and then find the ( 46^{\text{th}} ) term. use ( n ) represent the position of a term in the sequence, where ( n = 1 ) for the first term. (-90, -89, -88, -87, dots) (\boldsymbol{a_n = square}) (a_{46} = square)

Explanation:

Step1: Identify sequence type

This is an arithmetic sequence with first term \( a_1 = -90 \) and common difference \( d = 1 \) (since \(-89 - (-90) = 1\), \(-88 - (-89) = 1\), etc.).

Step2: Arithmetic sequence formula

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n - 1)d \). Substituting \( a_1 = -90 \) and \( d = 1 \), we get:
\( a_n = -90 + (n - 1)(1) \)
Simplify: \( a_n = -90 + n - 1 = n - 91 \)

Step3: Find the 46th term

Substitute \( n = 46 \) into \( a_n = n - 91 \):
\( a_{46} = 46 - 91 = -45 \)

Answer:

\( a_n = n - 91 \)
\( a_{46} = -45 \)